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An Introduction to Basic Functional Analysis AN INTRODUCTION TO BASIC FUNCTIONAL ANALYSIS T. S. S. R. K. RAO . 1. Introduction This is a write-up of the lectures given at the Advanced Instructional School (AIS) on ` Functional Analysis and Harmonic Analysis ' during the ¯rst week of July 2006. All of the material is standard and can be found in many basic text books on Functional Analysis. As the prerequisites, I take the liberty of assuming 1) Basic Metric space theory, including completeness and the Baire category theorem. 2) Basic Lebesgue measure theory and some abstract (σ-¯nite) measure theory in- cluding the Radon-Nikodym theorem and the completeness of Lp-spaces. I thank Professor A. Mangasuli for carefully proof reading these notes. 2. Banach spaces and Examples. Let X be a vector space over the real or complex scalar ¯eld. Let k:k : X ! R+ be a function such that (1) kxk = 0 , x = 0 (2) k®xk = j®jkxk (3) kx + yk · kxk + kyk for all x; y 2 X and scalars ®. Such a function is called a norm on X and (X; k:k) is called a normed linear space. Most often we will be working with only one speci¯c k:k on any given vector space X thus we omit writing k:k and simply say that X is a normed linear space. It is an easy exercise to show that d(x; y) = kx ¡ yk de¯nes a metric on X and thus there is an associated notion of topology and convergence. If X is a normed linear space and Y ½ X is a subspace then by restricting the norm to Y , we can consider Y as a normed linear space. Clearly jkxk ¡ kykj · kx ¡ yk and hence for any sequence fxngn¸1 ½ X, xn ! x ) kxnk ! kxk. 1 2 RAO When the metric d is complete, i.e., every Cauchy sequence in X converges in X, we say that X is a Banach space. It is easy to show that the Euclidian spaces Rn and Cn are Banach spaces. A subspace Y ½ X is a Banach space if and only if it is a closed subset of X. Example 1. Let K be a compact Hausdor® space, let C(K) denote the vector space of complex-valued continuous functions on K. For any f 2 C(K), de¯ne kfk = supfjf(k)j : k 2 Kg. This is called the supremum norm and C(K) is a Banach space. More concretely, let ¢ = fz 2 C : jzj · 1g and ¡ = fz : jzj = 1g. Let A = ff 2 C(¢) : f is analytic in ¢og. It is easy to see that A is a closed subspace and hence a Banach space. On the other hand P = fp 2 C(¢) : p is a polynomial in zg is not a closed set and hence is not a Banach space. If one only considers a locally compact set K then important spaces associated with this are, C0(K) = ff : K ! C : f is continuous and vanishes at infinityg (recall that f vanishes at in¯n- ity if for every ² > 0 there is a compact set C ½ K such that jfj < ² outside C) and Cc(K) = ff : K ! C : f is continuous and compactly supportedg (recall that f is compactly supported if fk : f(k) 6= 0g¡ is a compact set). Both these spaces are equipped with the supremum norm. It is easy to see that C0(K) is a Banach space and is the completion (in the metric space sense) of the space Cc(K). Example 2. Let (­; A; ¹) be a σ-¯nite measure space (more concretely take ­ = R, A to be the σ-¯eld of Lebesgue measurable sets, ¹ to be the Lebesgue measure). In what follows let us take two measurable functions that are equal almost every where (a.e) with respect to the measure ¹ as `equal'. For R 1 · p < 1, let Lp(¹) = ff : ­ ! C; f is measurable and jfjp d¹ < 1g. R 1 For f 2 Lp(¹), let kfk = ( jfjp d¹) p . Then Lp(¹) is a Banach space. A measurable function f is said to be essentially bounded if there exist a ® > 0 such that ¹((jfj)¡1((®; 1]) = 0. ® is called an essential bound. Let L1(¹) = ff : ­ ! C; f is measurable and essentially boundedg. For f 2 L1(¹) de¯ne kfk = inff® : ® is an essential boundg. Then L1(¹) is a Banach space. AN INTRODUCTION TO BASIC FUNCTIONAL ANALYSIS 3 A speci¯c situation of the above example and an easy generalization leads to a procedure for generating new classes of Banach spaces. We will only consider countable index sets. Example 3. Let fXngn¸1 be a sequence of Banach spaces. Let 1 · p < 1. Q P1 p Let X = fx = fxng 2 Xn : 1 kxnk < 1g. For x 2 X, de¯ne P 1 1 p p kxk = ( 1 kxnk ) . Then X is a Banach space. Next let X = fx = fxng : supjxnj < 1g. De¯ne kxk = supjxnj. This is again called the supremum norm. This is again a Banach space. When all the Xn are taken as the scalar ¯eld, the corresponding space X is denoted by `p and `1 respectively. Other important sequence spaces that are Banach spaces are, c, the space of convergent sequences of scalars and its subspace c0 of sequences converging to 0. Both the spaces are equipped with the supremum norm. They can also be treated as special cases of Example 1 for the discrete space N and its one point compacti¯cation. Next example again gives a way of generating new Banach spaces from a given collection. Example 4. Let X be a Banach space, let Y ½ X be a closed subspace. Consider the quotient vector space XjY . Elements of this space are denoted by [x] = fx + y : y 2 Y g. ¼ : X ! XjY de¯ned by ¼(x) = [x] is called the canonical quotient map. De¯ne k[x]k = d(x; Y ) = inffkx + yk : y 2 Y g. Now XjY is a Banach space. 4 RAO 3. Operators on normed linear spaces, Hahn-Banach theorem Let X be a normed linear space. X1 = fx 2 X : kxk · 1g,the ball with center at 0 and radius 1 is called the closed unit ball. By dilating by an r > 0 and translating by an x0 2 X we get the ball B(x0; r) = x0 + rX1. A set A ½ X is said to be bounded if there exists a r > 0 such that A ½ rX1. Equivalently kxk · r for all x 2 A. Clearly any compact set and in particular any convergent sequence, is a bounded set. Let X; Y be normed linear spaces, let T : X ! Y be a linear map. T is said to be bounded if it maps bounded sets to bounded sets. Such a T is called a bounded operator. Using linearity of T , we have that T is bounded if and only if T (X1) is a bounded set if and only if there exists an r > 0 such that kT (x)k · rkxk for all x 2 X. It is easily seen that a linear map is bounded if and only if it is continuous at 0 and then using the continuity of the vector space operations w. r. t the norm, one has that any bounded linear map is continuous. Let X be a normed linear space and let Y ½ X be a closed subspace. Let ¼ : X ! XjY be the quotient map. Then k¼k = 1. The following remarks which are like starred exercises, are designed to illustrate the role of ¯nite dimensional spaces. For the Euclidean spaces, any linear map T : Cn ! Cm is continuous. A linear, one-to-one and onto map T : X ! Y is said to be an isomor- phism if both T and T ¡1 are bounded. T is an isomorphism if and only if there exists constants m; M > 0 such that mkxk · kT (x)k · Mkxk. When m = M = 1 we say that T is an isometry. Any ¯nite dimensional normed linear space is isomorphic to some Eu- clidean space. So it is easy to see that any ¯nite dimensional normed linear space is a Banach space. Let X be a normed linear space and Y ½ X a ¯nite dimensional space. Then Y is closed. We denote by L(X; Y ) the vector space of bounded linear operators. For T 2 L(X; Y ) de¯ne kT k = supfkT (x)k : x 2 X1g. This is a norm. Also for any x 2 X, kT (x)k · kT kkxk. AN INTRODUCTION TO BASIC FUNCTIONAL ANALYSIS 5 When Y is a Banach space, L(X; Y ) is a Banach space. For T 2 L(X; Y ) and S 2 L(Y; Z), S ± T 2 L(X; Z) and kS ± T k · kSkkT k. When Y is the scalar ¯eld, a linear map f : X ! C is called a linear functional. The space of continuous linear functionals is denoted by X¤. ¤ ¤ For f 2 X , kfk = supfjf(x)j : x 2 X1g. X is a Banach space. Let X be an in¯nite dimensional normed linear space. Using the presence of in¯nitely many independent unit vectors one can see that there exists a linear map F : X ! C that is not bounded. Thus it is not clear in an abstract situation how one exhibits non-trivial bounded operators or bounded linear functionals? Theorem 5. (Hahn-Banach theorem): Let X be a normed linear space and let M ½ X be a subspace. Let f 2 M ¤. There exists a g 2 X¤ such that f = g on M and kfk = kgk.
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